Worksheet: Combining Functions

In this worksheet, we will practice adding, subtracting, multiplying, or dividing two given functions to create a new function and identifying the domain of the new function.

Q1:

What is the domain of the quotient ๐‘“๐‘”, in terms of the domains of ๐‘“ and ๐‘”? Assume that both domains are subsets of the set of real numbers.

  • Athe difference between the domain of ๐‘“ and the domain of ๐‘”
  • Bthe intersection of the domain of ๐‘“ and the domain of ๐‘”
  • Cthe larger of the domain of ๐‘“ and the domain of ๐‘”
  • Dthe union of the domain of ๐‘“ and the domain of ๐‘”
  • Ethe intersection of the domain of ๐‘“ and the domain of 1๐‘”

Q2:

Determine the common domain of the functions ๐‘›(๐‘ฅ)=โˆ’7๐‘ฅโˆ’7๏Šง and ๐‘›(๐‘ฅ)=โˆ’8๐‘ฅโˆ’64๏Šจ๏Šจ.

  • A โ„ โˆ’ { 7 , 8 }
  • B โ„ โˆ’ { โˆ’ 8 , โˆ’ 7 }
  • C โ„ โˆ’ { โˆ’ 8 , โˆ’ 7 , 8 }
  • D โ„ โˆ’ { โˆ’ 8 , 7 , 8 }
  • E โ„ โˆ’ { โˆ’ 8 , 8 }

Q3:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๐‘ฅ+3๐‘ฅโˆ’4๏Šจ and ๐‘”(๐‘ฅ)=๐‘ฅ+3, determine the value of (๐‘“+๐‘”)(โˆ’4) if possible.

  • A โˆ’ 6 5
  • B โˆ’ 6
  • Cundefined
  • D โˆ’ 1

Q4:

Find the domain of the function ๐‘“(๐‘ฅ)=โˆš๐‘ฅ+3+โˆš๐‘ฅโˆ’7๏Žข.

  • A [ โˆ’ 3 , โˆž )
  • B [ 7 , โˆž )
  • C ( โˆ’ 3 , โˆž )
  • D [ 3 , โˆž )

Q5:

If ๐‘“(๐‘ฅ)=๐‘ฅ+1 and ๐‘”(๐‘ฅ)=๐‘ฅ+1๏Šจ, then find and fully simplify an expression for (๐‘“โ‹…๐‘”)(๐‘ฅ).

  • A ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ ๏Šจ
  • B ๐‘ฅ + ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ ๏Šจ
  • C ๐‘ฅ + ๐‘ฅ + 2 ๏Šจ
  • D ๐‘ฅ + ๐‘ฅ + 1 ๏Šฉ

Q6:

If ๐‘“โ„โ†’โ„: where ๐‘“(๐‘ฅ)=4๐‘ฅโˆ’4, and ๐‘”[โˆ’8,โˆ’2)โ†’โ„: where ๐‘”(๐‘ฅ)=5๐‘ฅ+5, find the value of (๐‘“+๐‘”)(5) if possible.

  • Aundefined
  • B46
  • C16
  • D36

Q7:

Given that ๐‘“โˆถโ„โ†’โ„๏Šฐ, where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’19, and ๐‘”โˆถ[โˆ’2,13]โ†’โ„, where ๐‘”(๐‘ฅ)=๐‘ฅโˆ’6, evaluate (๐‘“โ‹…๐‘”)(7).

  • A โˆ’ 7 7 4
  • B โˆ’ 2 4 0
  • C724
  • D โˆ’ 1 2

Q8:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅ+9๐‘ฅ+15๐‘ฅ+54๏Šจ and ๐‘”(๐‘ฅ)=๐‘ฅ+8, determine the value of (๐‘“โˆ’๐‘”)(โˆ’6) if possible.

  • A โˆ’ 2
  • Bundefined
  • C1
  • D โˆ’ 5 3

Q9:

Given that ๐‘›(๐‘ฅ)=๐‘ฅ+16๐‘ฅโˆ’8๏Šง, ๐‘›(๐‘ฅ)=9๐‘ฅ+144๐‘ฅโˆ’8๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)รท๐‘›(๐‘ฅ)๏Šง๏Šจ, determine ๐‘›(๐‘ฅ) in its simplest form.

  • A ๐‘› ( ๐‘ฅ ) = 1 9
  • B ๐‘› ( ๐‘ฅ ) = 9
  • C ๐‘› ( ๐‘ฅ ) = 2 9
  • D ๐‘› ( ๐‘ฅ ) = 1 6
  • E ๐‘› ( ๐‘ฅ ) = 1 1 6

Q10:

Given that ๐‘›(๐‘ฅ)=5๐‘ฅโˆ’825๐‘ฅโˆ’4รท25๐‘ฅโˆ’30๐‘ฅโˆ’16125๐‘ฅ+8๏Šง๏Šจ๏Šจ๏Šฉ, ๐‘›(๐‘ฅ)=25๐‘ฅโˆ’450๐‘ฅโˆ’20๐‘ฅ+8๏Šจ๏Šจ๏Šจ, and ๐‘›(๐‘ฅ)=๐‘›(๐‘ฅ)ร—๐‘›(๐‘ฅ)๏Šง๏Šจ, simplify the function ๐‘›, and determine its domain.

  • A ๐‘› = 2 , domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • B ๐‘› = 1 2 , domain =โ„โˆ’๏ฌโˆ’25,25๏ธ
  • C ๐‘› = 1 2 , domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • D ๐‘› = 5 2 , domain =โ„โˆ’๏ฌโˆ’25,25,85๏ธ
  • E ๐‘› = 2 , domain =โ„โˆ’๏ฌโˆ’25,25๏ธ

Q11:

Given that ๐‘“โˆถ(โˆ’โˆž,4)โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅ+5๏Šง๏Šงsuchthat and ๐‘“โˆถ(โˆ’8,6)โ†’โ„๐‘“(๐‘ฅ)=2๐‘ฅ+13๐‘ฅ+15,๏Šจ๏Šจ๏Šจsuchthat find ๏€ฝ๐‘“๐‘“๏‰(๐‘ฅ)๏Šจ๏Šง and state its domain.

  • A 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 4 ) โˆ’ { โˆ’ 5 }
  • B 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 6 )
  • C 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 4 ) โˆ’ { โˆ’ 5 }
  • D 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ 8 , 4 )
  • E 2 ๐‘ฅ + 3 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 4 )

Q12:

Given that ๐‘“โˆถโ„โ†’โ„๐‘“(๐‘ฅ)=โˆ’3๐‘ฅโˆ’4where and ๐‘”โˆถ(1,7)โ†’โ„๐‘”(๐‘ฅ)=โˆ’2๐‘ฅโˆ’4,where find the value of ๏€ฝ๐‘“๐‘”๏‰(โˆ’1) if possible.

  • Anot defined
  • B0
  • C โˆ’ 1
  • D 1 2

Q13:

Given that ๐‘“โˆถ(โˆ’โˆž,2]โŸถโ„๐‘“(๐‘ฅ)=๐‘ฅ+5๏Šง๏Šงsuchthat and ๐‘“โˆถ(โˆ’โˆž,โˆ’1)โŸถโ„๐‘“(๐‘ฅ)=2๐‘ฅโˆ’๐‘ฅโˆ’6,๏Šจ๏Šจ๏Šจsuchthat find ๏€ฝ๐‘“๐‘“๏‰(๐‘ฅ)๏Šจ๏Šง and state its domain.

  • A 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 )
  • B 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 2 ]
  • C 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ) โˆ’ { โˆ’ 5 }
  • D ๐‘ฅ + 5 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ) โˆ’ { โˆ’ 5 }
  • E 2 ๐‘ฅ โˆ’ ๐‘ฅ โˆ’ 6 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ โˆž , โˆ’ 1 ]

Q14:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๏ญ2๐‘ฅ+2๐‘ฅ<โˆ’3,๐‘ฅโˆ’4โˆ’3โ‰ค๐‘ฅ<0,ifif and ๐‘”(๐‘ฅ)=5๐‘ฅ determine the domain of the function ๏€ฝ๐‘”๐‘“๏‰.

  • A ( โˆ’ โˆž , โˆ’ 3 )
  • B [ โˆ’ 3 , 0 )
  • C โ„ โˆ’ { 0 }
  • D ( โˆ’ โˆž , 0 )

Q15:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+1, find the domain of the function (๐‘“+๐‘”).

  • A [ โˆ’ 1 , โˆž )
  • B [ 1 , โˆž )
  • C ( โˆ’ โˆž , โˆ’ 1 ]
  • D [ โˆ’ 1 , โˆž ) โˆ’ { 0 , 5 }
  • E โ„ โˆ’ { 0 , 5 }

Q16:

If ๐‘“โˆถ(โˆ’7,8]โ†’โ„๏Šง where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’2๏Šง, and ๐‘“โˆถ[โˆ’8,4]โ†’โ„๏Šจ where ๐‘“(๐‘ฅ)=4๐‘ฅ+8๐‘ฅ+3๏Šจ๏Šจ, find (๐‘“โˆ’๐‘“)(๐‘ฅ)๏Šจ๏Šง and the domain of (๐‘“โˆ’๐‘“)๏Šจ๏Šง.

  • A 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 4 ]
  • B 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 8 ]
  • C 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ [ โˆ’ 7 , 4 )
  • D 4 ๐‘ฅ + 7 ๐‘ฅ + 5 ๏Šจ , ๐‘ฅ โˆˆ [ โˆ’ 8 , 4 ]
  • E โˆ’ 4 ๐‘ฅ โˆ’ 7 ๐‘ฅ โˆ’ 5 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 7 , 4 ]

Q17:

If ๐‘“โˆถโ„โŸถโ„๏Šฐ where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’17, and ๐‘”โˆถ[โˆ’25,4]โŸถโ„ where ๐‘”(๐‘ฅ)=๐‘ฅโˆ’11, then find (๐‘“+๐‘”)(๐‘ฅ) and its domain.

  • A ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 1 7 , [ 0 , 4 ]
  • B ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 8 , [ 0 , 4 ]
  • C ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 1 1 , ( 0 , 4 ]
  • D ( ๐‘“ + ๐‘” ) ( ๐‘ฅ ) = 2 ๐‘ฅ โˆ’ 2 8 , ( 0 , 4 ]

Q18:

If ๐‘“โ„โ†’โ„๏Šง๏Šฑ: where ๐‘“(๐‘ฅ)=4๐‘ฅ+4๏Šง, and ๐‘“(โˆ’9,6]โ†’โ„๏Šจ: where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๏Šจ, find and fully simplify (๐‘“โˆ’๐‘“)(๐‘ฅ)๏Šง๏Šจ and determine the domain of (๐‘“โˆ’๐‘“)๏Šง๏Šจ.

  • A 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ [ โˆ’ 9 , 0 ]
  • B 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ โ„ ๏Šฑ
  • C 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 6 ]
  • D 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 0 )
  • E 3 ๐‘ฅ + 5 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 6 ]

Q19:

If ๐‘“โ„โ†’โ„๏Šง๏Šฑ: where ๐‘“(๐‘ฅ)=โˆ’๐‘ฅโˆ’1๏Šง, and ๐‘“(โˆ’9,1)โ†’โ„๏Šจ: where ๐‘“(๐‘ฅ)=5๐‘ฅโˆ’3๏Šจ, find (๐‘“+๐‘“)(๐‘ฅ)๏Šง๏Šจ and the domain of (๐‘“+๐‘“)๏Šง๏Šจ.

  • A 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ โˆž , 1 )
  • B 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ โ„ ๏Šฑ
  • C 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 0 )
  • D 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ [ โˆ’ 9 , 0 ]
  • E 4 ๐‘ฅ โˆ’ 4 , ๐‘ฅ โˆˆ ( โˆ’ 9 , 1 )

Q20:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’5๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+4, determine the domain of the function (๐‘“โ‹…๐‘”).

  • A [ โˆ’ 4 , โˆž )
  • B [ 4 , โˆž )
  • C [ โˆ’ 4 , โˆž ) โˆ’ { 0 , 5 }
  • D ( โˆ’ โˆž , โˆ’ 4 ]
  • E โ„ โˆ’ { 0 , 5 }

Q21:

Given that ๐‘“โ„โŸถโ„๐‘“(๐‘ฅ)=๐‘ฅโˆ’4๏Šง๏Šฐ๏Šง:suchthat and ๐‘“(โˆ’9,1]โŸถโ„๐‘“(๐‘ฅ)=5๐‘ฅโˆ’2,๏Šจ๏Šจ:suchthat find (๐‘“โ‹…๐‘“)(๐‘ฅ)๏Šง๏Šจ and state its domain.

  • A 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 ๏Šจ , ๐‘ฅ โˆˆ [ 0 , 1 )
  • B 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 9 , โˆž )
  • C 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 9 , 1 ]
  • D 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 ๏Šจ , ๐‘ฅ โˆˆ โ„ ๏Šฐ
  • E 5 ๐‘ฅ โˆ’ 2 2 ๐‘ฅ + 8 ๏Šจ , ๐‘ฅ โˆˆ ( 0 , 1 ]

Q22:

Given that ๐‘“โˆถ(3,6]โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅ+10๐‘ฅ+25๏Šง๏Šง๏Šจsuchthat and ๐‘“โˆถ(โˆ’1,9)โ†’โ„๐‘“(๐‘ฅ)=๐‘ฅโˆ’3,๏Šจ๏Šจsuchthat find (๐‘“โ‹…๐‘“)(๐‘ฅ)๏Šง๏Šจ and state its domain.

  • A ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( โˆ’ 1 , 9 )
  • B ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( 3 , 6 ]
  • C ๐‘ฅ + 7 ๐‘ฅ โˆ’ 5 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ [ 3 , 6 )
  • D ๐‘ฅ โˆ’ 3 0 ๐‘ฅ + 1 0 ๐‘ฅ โˆ’ 7 5 ๏Šฉ ๏Šจ , ๐‘ฅ โˆˆ ( 3 , 6 ]

Q23:

If ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๏ญ๐‘ฅ+50<๐‘ฅ<2,2๐‘ฅ+5๐‘ฅโ‰ฅ2,ifif and ๐‘”(๐‘ฅ)=๐‘ฅ, find the domain of the function (๐‘“โ‹…๐‘”).

  • A ( 0 , 2 )
  • B [ 2 , โˆž )
  • C ( 0 , โˆž ) โˆ’ { 2 }
  • D โ„
  • E ( 0 , โˆž )

Q24:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅ๏Šจ and ๐‘”(๐‘ฅ)=โˆš5โˆ’๐‘ฅ, find the value of ๏€ฝ๐‘“๐‘”๏‰(5) if possible.

  • A0
  • B35
  • Cundefined
  • D โˆ’ 3 5

Q25:

Given that ๐‘“ and ๐‘” are two real functions where ๐‘“(๐‘ฅ)=๐‘ฅโˆ’1๏Šจ and ๐‘”(๐‘ฅ)=โˆš๐‘ฅ+5, find the value of ๏€ฝ๐‘”๐‘“๏‰(โˆ’2) if possible.

  • A โˆš 3
  • B3
  • C โˆ’ โˆš 3 3
  • Dundefined
  • E โˆš 3 3

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